|author:||Nikolaos Fountoulakis and Colin McDiarmid|
|title:||Upper bounds on the non-3-colourability threshold of random graphs|
|keywords:||sparse random graphs, 3-colourability, thresholds|
We present a full analysis of the expected number of
`rigid' 3-colourings of a sparse random graph. This shows
that, if the average degree is at least 4.99, then as n
-> ∞ the expected number of such
colourings tends to 0 and so the probability that the
graph is 3-colourable tends to 0. (This result is tight,
in that with average degree 4.989 the expected number
tends to ∞.) This bound appears independently in
Kaporis et al: A Note on the Non-Colourability Threshold of a Random Graph. We then give a minor improvement, showing that the probability that the graph is 3-colourable tends to 0 if the average degree is at least 4.989. |
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|reference:||Nikolaos Fountoulakis and Colin McDiarmid (2002), Upper bounds on the non-3-colourability threshold of random graphs, Discrete Mathematics and Theoretical Computer Science 5, pp. 205-226|
|bibtex:||For a corresponding BibTeX entry, please consider our BibTeX-file.|
|ps.gz-source:||dm050114.ps.gz (79 K)|
|ps-source:||dm050114.ps (227 K)|
|pdf-source:||dm050114.pdf (204 K)|
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